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It was the first time that Poole had seen a genuine horizon since he
had come to Star City, and it was not quite as far away as he had
expected.... He used to be good at mental arithmetic--a rare
achievement even in his time, and probably much rarer now. The
formula to give the horizon distance was a simple one: the square
root of twice your height times the radius--the sort of thing you
never forgot, even if you wanted to...
--Arthur C. Clarke, 3001, Ballantine Books, New York, 1997, page 71
In the above passage, Frank Poole uses a formula to determine the
distance to the horizon given his height above the ground.
- Use algebraic notation to express the formula Poole is using.
- Beginning the diagram below, use one of the circle theorems to
derive your own formula. You will need to add some more elements to
- Compare your formula to Poole's; you will find that they do not
match. How are they different?
- When I was a boy it was possible to see the Atlantic Ocean from
the peak of Mt. Washington in New Hampshire. This mountain is 6288
feet high. How far away is the horizon? Express your answer in
Assume that the radius of the Earth is 4000 miles.
Use both your formula and Poole's formula and comment on the results. Why
does Poole's formula work so well, even though it is not correct?
Wed Apr 21 08:26:07 EDT 1999